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Math 682 WINTER 2011, List of lectures
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On this page I will post content of all lectures. All handouts also will be posted here.
Monday, January 3: Universal enveloping algebra. PBW theorem.
Wednesday, January 5: Proof of PBW theorem, I.
Friday, January 7: Proof of PBW theorem, II.
Monday, January 10: Free Lie algebras; generators and relations.
Serre's relations.
Wednesday, January 12: Lie algebra L_0 decomposes as a direct sum
of L_-, H, L_+.
Friday, January 14: Proof of Serre's theorem.
Wednesday, January 19: End of proof of Serre's theorem.
Friday, January 21: Standard cyclic modules.
Monday, January 24: Verma modules. Possible highest weights of finite
dimensional irreducible representations.
Wednesday, January 26: Existense theorem for finite dimensional
irreducible representations.
Friday, January 28: Set of weights of irreducible finite dimensional
representation. Characters.
Monday, January 31: Examples of representations of sl_3.
Wednesday, February 2: W-invariant functions on H.
Friday, February 4: Invariant functions on L.
Monday, February 7: Chevalley restriction theorem.
Wednesday, February 9: Harish-Chandra theorem.
Friday, February 11: Proof of Harish-Chandra theorem.
Monday, February 14: Composition series of Verma modules.
Wednesday, February 16: Weyl character formula.
Friday, February 18: Dimension formula. Examples in rank 2.
Monday, February 21: Examples in type A. Vandermonde determinant and
Schur functions.
Wednesday, February 23: Branching Law gl(n)->gl(n-1). Gelfand-Cetlin basis.
Semistandard tableaux.
Friday, February 25: Kostant multiplicity formula for branching. Proof of
branching law gl(n)->gl(n-1).
Monday, February 28: Tensor products and Steinberg's formula.
Wednesday, March 2: How to decompose tensor products: example in type G_2.
Friday, March 4: Some results on tensor product decompositions.
Monday, March 7: Schur-Weyl duality.
Wednesday, March 9: Glimpse at Invariant Theory.
Friday, March 11: Chevalley basis and Chevalley groups of adjoint type.
Please find Final Exam.
THE END